An important class of quantum codes are quantum stabilizer codes. Binary stabilizer codes were
introduced by Calderbank et al. [1]} and Gottesman [2]}. Each binary stabilizer code is a quaternary additive code (an additive subgroup of \(\mathbb {F}_4^n\) ) which is dual containing with respect to a certain trace inner product [1]}.
For more information about the structure of quantum codes and their
algebraic constructions we refer to
the recent survey [4]}.
In this paper, we only restrict our attention to \(\mathbb {F}_4\) -linear subspaces of \(\mathbb {F}_4^n\) , and the following theorem gives the connection between quaternary linear codes and quantum codes.
[1]}
Let \(C\)  be a linear \([n,k,d]\)  code over \(\mathbb {F}_4\)  such that \(C^{\bot _h}\subseteq C\) , where \(C^{\bot _h}\)  is the Hermitian dual of \(C\) . Then we can construct an \([[n, 2k-n,d^{\prime }]]\)  binary quantum code, where \(d^{\prime }\) 
is the minimum weight in \(C\setminus C^{\bot _h}\) .
If \(C= C^{\bot _h}\)  then \(d^{\prime }=d\) .

If the quantum code of Theorem \(\ref {quantum def}\)  has minimum distance \(d^{\prime }=d\) , then the code is called a pure quantum code.
